) of n by n matrices with determinant equal to 1 is simple for all odd n > 1, when it is isomorphic to the projective special linear group.
The first classification of simple Lie groups was by Wilhelm Killing, and this work was later perfected by Élie Cartan.
Unfortunately, there is no universally accepted definition of a simple Lie group.
Authors differ on whether a simple Lie group has to be connected, or on whether it is allowed to have a non-trivial center, or on whether
In this article the connected simple Lie groups with trivial center are listed.
An important technical point is that a simple Lie group may contain discrete normal subgroups.
It emerged in the course of classification of simple Lie groups that there exist also several exceptional possibilities not corresponding to any familiar geometry.
These exceptional groups account for many special examples and configurations in other branches of mathematics, as well as contemporary theoretical physics.
As a counterexample, the general linear group is neither simple, nor semisimple.
This is because multiples of the identity form a nontrivial normal subgroup, thus evading the definition.
A semisimple Lie group is a connected Lie group so that its only closed connected abelian normal subgroup is the trivial subgroup.
(Authors differ on whether the one-dimensional Lie algebra should be counted as simple.)
Over the complex numbers the semisimple Lie algebras are classified by their Dynkin diagrams, of types "ABCDEFG".
The different real forms correspond to the classes of automorphisms of order at most 2 of the complex Lie algebra.
(For example, the universal cover of a real projective plane is a sphere.)
The irreducible simply connected symmetric spaces are the real line, and exactly two symmetric spaces corresponding to each non-compact simple Lie group G, one compact and one non-compact.
A symmetric space with a compatible complex structure is called Hermitian.
The compact simply connected irreducible Hermitian symmetric spaces fall into 4 infinite families with 2 exceptional ones left over, and each has a non-compact dual.
In addition the complex plane is also a Hermitian symmetric space; this gives the complete list of irreducible Hermitian symmetric spaces.
In the symbols such as E6−26 for the exceptional groups, the exponent −26 is the signature of an invariant symmetric bilinear form that is negative definite on the maximal compact subgroup.
It turns out that the simply connected Lie group in these cases is also compact.
Compact Lie groups have a particularly tractable representation theory because of the Peter–Weyl theorem.
Just like simple complex Lie algebras, centerless compact Lie groups are classified by Dynkin diagrams (first classified by Wilhelm Killing and Élie Cartan).
For the infinite (A, B, C, D) series of Dynkin diagrams, a connected compact Lie group associated to each Dynkin diagram can be explicitly described as a matrix group, with the corresponding centerless compact Lie group described as the quotient by a subgroup of scalar matrices.
As with the B series, SO(2r) is not simply connected; its universal cover is again the spin group, but the latter again has a center (cf.
The diagram D2 is two isolated nodes, the same as A1 ∪ A1, and this coincidence corresponds to the covering map homomorphism from SU(2) × SU(2) to SO(4) given by quaternion multiplication; see quaternions and spatial rotation.
Also, the diagram D3 is the same as A3, corresponding to a covering map homomorphism from SU(4) to SO(6).
In addition to the four families Ai, Bi, Ci, and Di above, there are five so-called exceptional Dynkin diagrams G2, F4, E6, E7, and E8; these exceptional Dynkin diagrams also have associated simply connected and centerless compact groups.
A simply laced group is a Lie group whose Dynkin diagram only contain simple links, and therefore all the nonzero roots of the corresponding Lie algebra have the same length.